Mechanical metamaterials made of freestanding quasi-BCC nanolattices of gold and copper with ultra-high energy absorption capacity

Nanolattices exhibit attractive mechanical properties such as high strength, high specific strength, and high energy absorption. However, at present, such materials cannot achieve effective fusion of the above properties and scalable production, which hinders their applications in energy conversion and other fields. Herein, we report gold and copper quasi-body centered cubic (quasi-BCC) nanolattices with the diameter of the nanobeams as small as 34 nm. We show that the compressive yield strengths of quasi-BCC nanolattices even exceed those of their bulk counterparts, despite their relative densities below 0.5. Simultaneously, these quasi-BCC nanolattices exhibit ultrahigh energy absorption capacities, i.e., 100 ± 6 MJ m−3 for gold quasi-BCC nanolattice and 110 ± 10 MJ m−3 for copper quasi-BCC nanolattice. Finite element simulations and theoretical calculations reveal that the deformation of quasi-BCC nanolattice is dominated by nanobeam bending. And the anomalous energy absorption capacities substantially stem from the synergy of the naturally high mechanical strength and plasticity of metals, the size reduction-induced mechanical enhancement, and the quasi-BCC nanolattice architecture. Since the sample size can be scaled up to macroscale at high efficiency and affordable cost, the quasi-BCC nanolattices with ultrahigh energy absorption capacity reported in this work may find great potentials in heat transfer, electric conduction, catalysis applications.


Supplementary Discussions
Supplementary Discussion 1 The microstructure and the purity of the gold and copper quasi-

BCC nanolattices
To examine the microstructure and the purity of our gold and copper quasi-BCC nanolattices, we have performed all the following characterizations, i.e., X-ray diffraction (XRD), high-resolution TEM, backscattered electron SEM, energy disperse x-ray spectra (EDS), and electron energy loss spectroscopy (EELS). Benefiting from the above methods, we have determined that microstructures of our gold and copper quasi-BCC nanolattices are polycrystalline and, within the detection limit of the above techniques, the nanolattices are in high purity and no impurity was detected. Details are illustrated below.
The microstructures of the gold and the copper quasi-BCC nanolattices were evaluated by XRD and TEM. XRD data of either gold or copper quasi-BCC nanolattices exhibit typical polycrystallinelike patterns of face-centered cubic (FCC) phases, where the peaks of main crystal planes appear and, in each pattern, the (111) crystal plane shows the strongest diffraction intensity (Supplementary Fig.   1). We have also verified the polycrystalline microstructure by TEM, taking gold as an example ( Supplementary Fig. 2). It is seen that two beams are composed of three grains, confirming the polycrystalline microstructure.
The purity of the gold and the copper were examined by backscattered electron SEM (BSE-SEM), EDS, and EELS techniques. The BSE-SEM images of gold and copper quasi-BCC nanolattices have similar morphologies as those of the secondary electron SEM (SE-SEM) images ( Supplementary Fig.   3). Although the BSE-SEM images have lower signal-to-noise ratio, there is no observable contrast difference, reflecting the nanolattices have high purities at the microscale. This observation is further 5 supported by the EDS analysis ( Supplementary Fig. 4). The EDS data taken from different regions illustrate that our nanolattices are only composed of pure gold or copper, respectively. The high purity of gold and copper quasi-BCC nanolattices are further consolidated by EELS spectra ( Supplementary   Fig. 5), where no other elements were detected. Based on the above results, it is safe to conclude that our nanolattices are in high purity.

Supplementary Discussion 2 Factors affecting the mechanical properties of the gold and copper quasi-BCC nanolattices
Effect of offset nodes on mechanical properties of gold and copper quasi-BCC nanolattices In this work, the influence of offset nodes in the quasi-BCC nanolattice on its mechanical properties was studied through finite element simulation. Combined with the previous mechanical test results of gold pillars, gold nanowires, and nanoporous gold 1-7 , the mechanical parameters of gold materials with size effects were set as shown in Supplementary At the relative density of 0.20, the quasi-BCC nanolattice has a 53% decrease, i.e., from 26.9 MPa for 6 the periodic nanolattice to 12.7 MPa for the quasi-BCC nanolattice. To sum up, the node offset effects have a greater impact on the mechanical properties of our quasi-BCC nanolattices, as comparing with those on the reported octet-truss nanolattices (Ref. 21).
Effects of surface roughness on mechanical properties of gold and copper quasi-BCC nanolattices Surface roughness, beam waviness, misalignment of nodes, and others are factors that influence the measured stiffness of a metamaterial. In our quasi-BCC nanolattices, from the SEM image of a FIB-milled pillar ( Supplementary Fig. 8), it is clearly seen that some neighboring protrusions (beam ends) are in different heights and, as a result, form surface roughness. It would be great to give a relative degree of the surface roughness. Unfortunately, unlike the surface roughness of a nonporous solid material which can be quantitatively evaluated by techniques such as atomic force microscopy, the relative degree of the surface roughness of porous materials, in particular stochastic truss porous materials like ours, are hardly evaluated quantitatively. For the beam waviness, we determined the waviness from a TEM image ( Supplementary Fig. 9). The results show that the beam waviness of beam 1 is 0.5 nm and the beam waviness of beam 2 is 0.6 nm, respectively, both are below 1 nm and less than 1% (beam diameter 69±2 nm). Thus, we think beam waviness should play a minor role in influencing the stiffness, as comparing with the surface roughness.
To illustrate the effects of contact state and surface roughness on the stiffness and the strength, we have performed additional finite element simulations ( Supplementary Fig. 10). The areal density and the beam diameter of the simulated quasi-BCC nanolattices are 7.1×10 8 ×4 cm -2 and 69 nm, respectively, corresponding to those of the sample Au-69. The model volume is 1×1×1 μm 3 . Surface contact state 1 represents the highest contact level (contact area 190249 nm 2 ), namely, the upper ends of beams are fully in contact with the indenter, which is the case involved in this paper. The state 2 7 (contact area 98358 nm 2 ) and the state 3 (contact area 34620 nm 2 ) represent that the upper ends partly contact with the indenter with reduced contact level. In the state 4, about a half of the number of beams are in contact with the indenter at the contact level prescribed in the state 3, and the rest do not contact.
Compared to the state 3, the state 4 has surface roughness. One can see that the compressive stiffness degrades from 541.0 MPa to 332.4 MPa, as the contact changes from the state 1 to the state 3 ( Supplementary Fig. 10e). In comparison, the strength degrades from 12.7 MPa to 12.4 MPa, reflecting that the strength is insensitive to the contact state ( Supplementary Fig. 10f). In short, the surface roughness has an obvious impact on the stiffness and a limited influence on the strength.
Complementary to the above simulations, the experimental stress-strain curve may also give some hints about the surface roughness. We found the stress-strain curves of all the samples will go through two parts at the initial stage during compression. The first part has a smaller slope E1 and the second part has a larger slope E2.  Fig. 11b). We think the smaller slope (E1) of the first part is very likely due to the surface roughness and the larger slope (E2) is the stiffness of our quasi-BCC nanolattices. This explanation may be supported by the findings reported previously 8,9 . As such, the turning point of strain reflects, to some extent, the surface roughness.
In short, both the contact level and the surface roughness have obvious impact on the stiffness and limited influence on the strength. 8

quasi-BCC nanolattices
To quantitatively evaluate the effect of nodal offsets on nanolattice mechanical properties, we reviewed theoretical models for ideal cell compressive modulus and strength, whose equivalent compressive modulus for body-centered cubic (BCC) structures are 10,11 : where E is the Young's modulus of the parent material. The compressive strength of the BCC structure can be expressed as 10,11 : where σy is the yield strength of the parent material, and σcr is the ultimate strength of the bracing in the buckling state, which can be expressed as 10 where the factor k=1 is determined by the zero rotational stiffness of the end nodes of the nanobeams, I is the moment inertia of the beam (=d 4 /64), and A is the cross sectional area of the circular beam (d 2 /4). In above equation, Et denotes the tangent modulus of the parent material, which depends on the strain hardening characteristics, typically one-tenth of the Young's modulus. At higher relative densities (~0.5), the compressive strength of the nanolattice is governed by the yield strength σy of the material. However, as the relative density decreases (<0.3), the structs become more and more slender, and under certain conditions the compressive strength of the nanolattice is governed by 9 buckling rather than yielding. The compressive strength in this case is obtained by replacing σcr in equation 2 with the buckling strength of the truss member. In this way, we obtained the quantitative value of the compressive strength of periodic body-centered cubic structure through finite element simulation and theoretical calculation, and the two are highly consistent, the relative error is less than 5%. For the metallic quasi-BCC nanolattices in this paper, when the feature size and relative density are fixed, it is only different from the periodic body-centered cubic structure in structure. Or we can consider quasi-BCC nanolattices as nanolattice structures with full-node offset defects. Its compressive strength could be given by: Where η is the structure factor, which is numerically equal to 0.5, and we can accurately calculate the compressive strength of the quasi-BCC nanolattice by Equation (4), see details in Supplementary   Table 4. The high agreement between the above theoretical calculation results and the experimental results of gold and copper quasi-BCC nanolattice shows the generality of the theory.

BCC nanolattices
For a single monolithic quasi-BCC nanolattice, the lower density limit is basically determined by the connectivity of beams and, in turn, by the areal density, the diameter, and the length of nanobeams (a longer beam has more possibilities to connect to other beams). To elaborate on the lower limit of the relative density, we have carried out the analysis with the aid of geometric models and finite element simulations ( Supplementary Fig. 12) in combination with experimental test ( Supplementary   Fig. 13), using gold as the material. The analysis of geometric models and finite element simulations 10 suggest that, the quasi-BCC nanolattice of the relative density of only 0.01 nearly remains structural integrity, given that it is in the same dimensions of FIB milled pillars, i.e., 10×10×10 µm 3 ( Supplementary Fig. 12). However, because of the morphological damage induced by the surface tension of dichloromethane during dissolving polycarbonate template, the experimental test shows that the quasi-BCC nanolattice with a relative density of 0.15 starts to lose its structural integrity partially ( Supplementary Fig. 13). Details are elucidated below.
The analyses of geometric models and finite element simulations were carried out to estimate the lower limit of relative density from the perspective of structural integrity and performance degradation, respectively. For the analyses, the areal density was fixed to be 7. In the case of model volume of 1×1×1 µm 3 , 25% beams fail to connect to any other beams, namely, the nanolattice loses its partial structural integrity ( Supplementary Fig. 12a Fig. 12d). Moreover, it is seen that the stiffness and the strength are highly dependent on the relative density ( Supplementary Fig. 12e,f). In summary, the relative density as low as 0.01 nearly keeps the structural integrity and yields certain mechanical strength.
We have also searched the lower limit of the relative density under our experimental conditions.
We found that the gold quasi-BCC nanolattice with a relative density of 0.15 starts to lose its structural integrity partially, which is attributed to the morphological damage induced by the surface tension of dichloromethane solvent during dissolving polycarbonate template. For this sample, the area density is 7.1×10 8 ×4 cm -2 (consistent with the sample Au-117), and the beam diameter is 60±3 nm. The SEM images of the quasi-BCC nanolattices are shown in Supplementary Fig. 13. It is seen that, although the sample keeps monolithic ( Supplementary Fig. 13a), the surface morphology is partially damaged at the microscopic scale ( Supplementary Fig. 13b). Guided by the analyses of geometric models and finite element simulations, it is reasonable to anticipate the quasi-BCC nanolattices with lower relative densities below 0.2 could be experimentally fabricated by further refining the experimental process, for example, reducing or eliminating surface tension of solvents using a freeze-drying method 12 .